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Actually, the total raised was $731.83. All money raised will be donated to the March of Dimes, so no robbery involved. Actually, while we were assembling, students were coming up and donating even more money from the consoles and glove boxes of their cars. We're probably easily over $750 by now. And yes, it's Eudora, Kansas.

So, the deal with the Sierpinski triangle is that it is composed of three exact copies of itself, stacked on top of each other in a triangle formation. This mathematical ideal is obviously not possible in the real world since you would need an infinite number of infinitesimally small pennies, so the smallest possible version of a real world "pseudo-Sierpinski" triangle is just one single penny.

The next level of that would be three copies of that single penny in a triangle. The next level after that would be three copies of that triangle stacked in a triangle (i.e. 9 pennies). The next level after that would be three copies of that 9-penny triangle, which would be 9*3 = 27 pennies.

Obviously, for each level up you go, the number of pennies is multiplied by 3. So lets say that the simplest triangle, that with only 1 penny (lets call it the "degenerate Sierpinski triangle", which is a word math people like to use) is "level 0". Then "level 1" is three pennies, "level 2" is 9 pennies, "level 3" is 27 pennies, and so on. Clearly, for "level x", the number of pennies needed is 3*3*3*...*3, i.e. 3 multiplied by itself x times. In math terms, this is "3 to the power of x", or 3x

Now, since we know that roughly 60,000 pennies were used, in order to find out the what level Sierpinski triangle it was, we have to solve this equation:

3x = 60000

This is easily done by taking the logarithm of both sides:

log( 3x ) = log(60000)
x * log(3) = log(60000)
x = log(60000)/log(3)

(this is the same as taking the logarithm of 60000 to the base 3). Lets plug that into a calculator and see what happens. Look at that, you get almost exactly 10! So we can surmise that the number of levels used to build it was 10, and that the exact number of pennies (if they built it perfectly) should be 310 which is equal to 59,049.

Now then, what is the height? Well, since for each level you go up, the height (as well as the width) doubles, we can say that if the diameter of one penny is equal to "p" and the number of levels is equal to "x", the height should be:

height = p * 2x

Now, wolfram alpha tells me that the diameter of one penny is 19mm, and since the thing had ten levels, the total height (if they built it perfectly) should be:

height = 19 mm * 210 = 19,456 mm = 19.456 meters = 63.83 feet

This isn't the exact answer to the problem you asked, but I'll leave it as an exercise to the reader to try and figure out how to go "the other way", i.e. calculating the number of pennies from the height.

TL;DR: Math is awesome!

(PS. This is actually kinda boring, it's much more fascinating (and pretty much as simple) to analyze the area of a Sierpinski triangle. If you do that, you realize that it's actually not a two-dimensional object, it's actually a roughly 1.585 dimensional object. But this comment is already too long)

EDIT: by the way, because the pictures are nice and zoomed in (especially the second one), you can actually count the number of levels used, and it is, indeed, 10 levels. So the math checks out!

I agree that math is awesome, and I'm impressed both with the amount of effort in your response and my inability to verify it ;-)

From my perspective as a software engineer, I was thinking this:
Let d be the diameter of an elementary circle used to create a structure in the general form of a Sierpinski triangle and let h equal the height of a given equilateral triangle. Implement a function that creates the largest complete Sierpinski triangle out of circular elements of diameter d within the bounds of height h and return the number of circular elements used.

I spent about 15 minutes working this out before I flipped my desk over and made myself a drink.

It's the exact same math you use. You just figure out the size of the smallest Sierpinski triangle, then try and see how big x can be such that size * 2x is less than or equal to the size of the bigger triangle. Then if you want the number of pennies you take 3x

(I wrote a much longer and more detailed explanation with lots of irrelevant geometry, but I deleted it because when I read it, I put myself to sleep :) Suffice to say, what I did up that is what you need to do)

*100

60,000

If you didn't know, if you're multiplying easy numbers, say 13 and 200, just multiply 13 and 2 which gives you 26 then add two zero's (2,600) that you left out of the math and reintroduce them to basically simplify the math by having fewer numbers to deal with in your head.

It didn't have to be, but the math teacher in charge was out all day, it was his class for the day, although I just skipped all my classes and helped all day. It's nice to go to a small school some times.

I would consider this art, in fact it is quite inspiring. I wish more people would do things like this, your teacher is an amazing person for setting this up. Even if the record is broken you will have been part of something epic. I wish you were closer so I could see it in person!

wow you guys have lots of patience, prob should have put a tarp or something underneath for easy clean up though ha. I am imagining fingernails peeling away from the skin as they get snagged while clawing the pavement

This is obviously the best use anyone has come up with for pennies. I wish something like this happened when I was in high school. Don't get me wrong, I had an awesome math teacher, but this is just amazing.